
Recalling that \(P_1^a(i)\ge \frac{\delta }{c_4}\)  and by Lemma REF ,
we can set \(P_1(t,i)=\lim \limits _{a\rightarrow 0}P_1^{a}(t,i)\)  for all \(i\in \mathcal {M}\) . Note that we can regard
\((P_1^a(i),\Lambda _1^a(i))\)  as the solution of a scalar-valued quadratic BSDE on \([0,T]\)  for each \(i\in \mathcal {M}\) . Let \(a\rightarrow 0\) , by Proposition 2.4 in [1]}, there exists a process \(\Lambda _1(i)\in L^2_{\mathcal {F}^W}(0,T;\mathbb {R}^n)\)  such that \((P_1(i),\Lambda _1(i))\)  is a solution to ESRE (REF ) on the time interval \([0,T]\)  with terminal value \(P_1(T,i)\) . Similar argument as in Theorem REF  yields that \(\Lambda _1(i)\)  is independent of \(T\) . Thus \((P_1(i),\Lambda _1(i))_{i\in \mathcal {M}}\)  is a uniformly positive solution to (REF ).
\(\Box \)
